A hyperbola has a vertical transverse axis of length 6 and asymptotes of y = (2x/5)+2 and y = (-2x/5)-4.
I am having trouble finding the focal length. I mean I think I've found it, but apparently it's incorrect.
c = sqrt(a^2+b^2) = sqrt(5^2+2^2) = sqrt(29)
--> By the way, the eccentricity e = c/a = (sqrt(29))/2 = 2.69, which is correct. So I suppose a = 2 and b = 5.
focal length = 2c = 2(sqrt(29)) = 10.77 (supposedly incorrect)
Regardless, what about the focal length I obtained? Is it correct?
To find the focal length of a hyperbola with a vertical transverse axis, you can use the formula:
focal length = √(a^2 + b^2)
where a is the distance from the center to the vertex and b is the distance from the center to the asymptote.
In your case, you correctly determined that a = 2 and b = 5. However, there seems to be a misinterpretation of the given asymptotes. Let's double-check the slopes of the asymptotes to ensure accurate calculations.
The slope of a line in the form y = mx + c is given by m, where m is the coefficient of x. Comparing the given asymptotes:
Asymptote 1: y = (2x/5) + 2
Slope = 2/5
Asymptote 2: y = (-2x/5) - 4
Slope = -2/5
Now we can find the focal length using the formula:
focal length = √(a^2 + b^2)
= √(2^2 + 5^2)
= √(4 + 25)
= √29
The correct focal length for the hyperbola is √29 or approximately 5.39.
To find the focal length of a hyperbola given its equation and asymptotes, you can follow these steps:
1. Write the equation of the hyperbola using the given information. In this case, the equation of the hyperbola can be written in the standard form as (y-k)^2 / a^2 - (x-h)^2 / b^2 = 1, where (h,k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the co-vertices.
2. Based on the given information, we know that the hyperbola has a vertical transverse axis of length 6. This means that the value of a is 6/2 = 3.
3. The asymptotes of the hyperbola are given as y = (2x/5) + 2 and y = (-2x/5) - 4. To find the value of b, we need to relate it to the slope of the asymptotes. For a hyperbola in standard form, the slopes of the asymptotes are given by b/a and -b/a. From the given asymptotes, we can determine that b = 5 (since |2/5| = |(-2)/5| = b/a = 5/3).
4. Now we can calculate the focal length of the hyperbola using the formula c = sqrt(a^2 + b^2). In this case, c = sqrt(3^2 + 5^2) = sqrt(9 + 25) = sqrt(34).
5. Finally, the focal length of the hyperbola is 2c, which is 2 * sqrt(34) = 2sqrt(34) ≈ 11.66 (rounded to two decimal places).
So, the correct value for the focal length of the given hyperbola is approximately 11.66, rather than 10.77.